Lesson plan Grade 12 Math
Norbourne Jar Test: Comparing Two Coagulants with Simulation
A 55-minute grade 12 lesson using a fictional plant's jar test to compare two coagulants, then a randomization simulation to judge whether the difference is real.
Objectives
- Students will compute and compare mean settled turbidity for two randomly assigned treatments.
- Students will use a simulated randomization distribution to judge whether a difference is explainable by chance.
- Students will classify a water treatment study as an experiment and explain the role of random assignment.
Vocabulary
- Jar test
- A jar test is a small-scale experiment operators run in the lab before treating a whole tank of water, testing different amounts of chemical in separate jars to see which dose clears the water best. It helps operators avoid wasting chemicals or under-treating the water.
Also: coagulant, settled turbidity, random assignment, randomization distribution, removal efficiency
Teacher overview
Before a treatment plant switches coagulants, it runs a jar test: identical jars of the same raw water, dosed differently, stirred and settled under matched conditions. This lesson gives students the results of one such test at the fictional Norbourne water plant, where twelve jars were split at random between the plant's current alum dose and a polymer-aided blend. Students compute the difference in mean settled turbidity, then use a simulated randomization distribution to decide whether a difference that size is surprising. The misconception this lesson corrects is the belief that any difference in means settles the question, when the size of a difference says nothing on its own about how often chance alone would produce one at least that large.
Procedure
Framing the pilot (8 min): Describe the decision facing the fictional Norbourne plant: the chemical supplier claims a polymer-aided blend settles raw water more effectively than the plant's current alum dose. Explain that the operators drew one large batch of raw water, filled twelve identical jars, and assigned six jars to each coagulant by drawing numbered cards. Ask the class why the assignment was randomized rather than left to the operator's choice, and record two or three answers on the board.
Teacher-modeled difference of means (10 min): Work a two-jar version at the board so students see the arithmetic before they own it. Suppose one jar dosed with alum settles to 3.0 NTU and one jar dosed with the blend settles to 2.2 NTU. Model the subtraction, then ask whether a single pair could settle the question, and draw out that two jars cannot separate the coagulant's effect from ordinary jar-to-jar variation. Do not use the handout's twelve jar values in this demonstration.
Handout items 1 through 5 (12 min): Students work independently or in pairs on the two means, the difference, and the two removal efficiencies. Circulate and check that students subtract in a consistent direction, alum minus blend, so the sign of the difference stays meaningful.
Randomization simulation, teacher-led (10 min): Write the twelve settled-turbidity values on index cards. Shuffle all twelve, deal six to each pile, and compute the difference in means for that shuffle in front of the class. Repeat twice more, recording each result on the board, and point out that the coagulant labels no longer mean anything after shuffling. Explain that the handout's table reports 40 such shuffles run by computer, and that the question is how often chance alone puts a difference at least as large as the observed one.
Handout items 6 through 10 (10 min): Students count from the table, convert the count to a proportion, and write their interpretation. Remind them that the proportion answers a specific question about chance, not about whether the blend is worth its price.
Formative check and close (5 min): Poll the class on one statement: "A proportion of 0.05 means the blend is guaranteed to work better." Ask for thumbs up or down. Move on only when at least 80 percent correctly reject the statement; if fewer do, work item 8 together before dismissal.
Student Handout
All values below describe a jar test at the fictional Norbourne water plant. Twelve jars were filled from one batch of raw water measuring 14 NTU, then assigned at random, six jars to each coagulant. Each jar was stirred, then allowed to settle for 30 minutes before its turbidity was read.
Settled turbidity, in NTU:
| Reading | Coagulant A (alum) | Coagulant B (blend) |
|---|---|---|
| 1 | 1.8 | 1.3 |
| 2 | 2.1 | 1.5 |
| 3 | 1.6 | 1.1 |
| 4 | 2.4 | 1.8 |
| 5 | 1.9 | 1.4 |
| 6 | 2.2 | 1.7 |
The readings are listed in the order the jars were read. The six jars in each column are a group, not a matched pair with the row beside them.
A computer then reshuffled all twelve readings into two groups of six, 40 times, and recorded the difference in group means each time. Those 40 simulated differences are summarized here:
| Simulated difference, A minus B (NTU) | Number of shuffles |
|---|---|
| -0.53 to -0.41 | 2 |
| -0.40 to -0.28 | 5 |
| -0.27 to -0.15 | 7 |
| -0.14 to -0.02 | 6 |
| -0.01 to 0.11 | 6 |
| 0.12 to 0.24 | 7 |
| 0.25 to 0.37 | 4 |
| 0.38 to 0.52 | 1 |
| 0.53 to 0.66 | 2 |
Find the mean settled turbidity for Coagulant A, in NTU. Round to the nearest hundredth.
Find the mean settled turbidity for Coagulant B, in NTU. Round to the nearest hundredth.
Find the observed difference in means, A minus B, in NTU. Round to the nearest hundredth.
The raw water measured 14 NTU. Find Coagulant A's removal efficiency, where removal efficiency equals influent minus effluent, divided by influent, times 100. Round to the nearest tenth of a percent.
Find Coagulant B's removal efficiency the same way. Round to the nearest tenth of a percent.
Using the table of 40 shuffles, how many simulated differences are at least as large as your problem 3 answer?
Write your problem 6 count as a proportion of the 40 shuffles.
Using your problem 7 proportion, write two or three sentences on whether chance alone explains the observed difference comfortably.
Is this jar test a sample survey, an observational study, or an experiment? Name it, and explain in one or two sentences what assigning the jars at random accomplishes.
After filtration, both coagulants produced finished water at or below the 0.3 NTU standard. Explain in two or three sentences why the plant might still care about the settled-water difference measured here.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: 2.00 NTU.
The six alum readings total 12.0 NTU, and 12.0 / 6 = 2.00 NTU.Answer: 1.47 NTU.
The six blend readings total 8.8 NTU, and 8.8 / 6 = 1.4667, which rounds to 1.47 NTU.Answer: 0.53 NTU.
2.00 - 1.47 = 0.53 NTU. The blend settled lower, so the difference is positive in the stated direction.Answer: 85.7 percent.
(14 - 2.00) / 14 x 100 = 12.0 / 14 x 100 = 85.71, which rounds to 85.7 percent.Answer: 89.5 percent.
(14 - 1.4667) / 14 x 100 = 12.5333 / 14 x 100 = 89.52, which rounds to 89.5 percent, about 3.8 percentage points better than alum.Answer: 2 shuffles.
Only the final row of the table, covering 0.53 to 0.66 NTU, reaches the observed difference of 0.53, and that row holds 2 shuffles.Answer: 0.05.
2 / 40 = 0.05, or 5 percent of the shuffles.Answer: chance is an uncomfortable explanation.
Accept any answer that reads the proportion correctly. When the labels are shuffled at random, a difference of 0.53 NTU or larger showed up in only 2 of 40 tries, so chance alone rarely produces a gap this size. That makes the coagulant a more plausible explanation than random jar-to-jar variation, though 40 shuffles is a small simulation and the result is close to the usual 5 percent benchmark.Answer: an experiment.
The operators assigned the treatment rather than observing whichever coagulant a jar happened to get, which makes it an experiment. Random assignment spreads unmeasured differences between jars evenly across the two groups, so the difference in settled turbidity can reasonably be attributed to the coagulant rather than to some jar-specific quirk.Answer: lower settled turbidity protects the filters.
Accept any well-supported answer. Settled water that arrives at the filters cleaner leaves less solids for the filters to capture, which lengthens filter runs, reduces backwash water and pumping cost, and leaves more margin before the 0.3 NTU standard is at risk during a storm. Meeting the standard today does not mean the plant has room to spare when raw water quality gets worse.
Differentiation
Support
Give students the two column totals for the fictional Norbourne jar data, 12.0 and 8.8, so items 1 and 2 are single divisions, and provide the removal efficiency formula already substituted for item 4. Pair students for items 6 and 7, and provide a highlighted copy of the shuffle table with the rows at or above 0.53 already outlined.
Extension
Ask students to state what the fictional plant would need to see to be more confident: have them describe how the proportion in item 7 would likely change with 400 shuffles instead of 40, and design a follow-up jar test with a larger number of jars, naming what stays randomized and what stays fixed.
ELL supports
Pre-teach "jar test," "coagulant," "settled," and "shuffle" with a labeled photo or sketch of six jars on a stirrer, and post the sentence frame "A difference of ___ NTU or larger happened in ___ out of 40 shuffles, so ___." Allow students to answer items 8 through 10 with a short frame-supported sentence rather than a full paragraph.
Extensions
Have students bring the cost side into the fictional Norbourne decision. Give teams a supplier price for each coagulant and the plant's daily flow, and ask them to compute the added annual chemical cost of the blend, then argue in a short memo whether the turbidity gain justifies it. The memo forces students to say plainly what the statistics do and do not establish, which is the same distinction a plant superintendent has to make in front of a utility board.